Question: 2(3w) + 2(w), the sum of twice the length and twice the width
Expression 3:
2(3w + w), twice the sum of the length and the width
Solution 3w + w + 3w + w; 2(3w) + 2(w); 2(3w + w)
Pair/Share
How could you use the
distributive property to
write one of the
expressions for the
perimeter?
16 The expression 12c 6 18 represents the perimeter of a regular hexagon.
Write two different expressions to describe its perimeter. Then write an
expression for the length of one of its sides.
Show your work.
The sides of a regular
polygon are all the
same length.
Solution
Pair/Share
How might drawing a
diagram help you solve
the problem?
1. **State the problem:** We are given expressions involving length and width, and asked to use the distributive property to write expressions for the perimeter. Also, for a regular hexagon with perimeter expression $$12c - 18$$, find two expressions for the perimeter and the length of one side.
2. **Recall the distributive property:** $$a(b + c) = ab + ac$$. This allows us to rewrite sums inside parentheses multiplied by a number.
3. **Analyze the expressions:**
- Expression 1: $$2(3w) + 2(w)$$ means twice the length plus twice the width.
- Expression 2: $$2(3w + w)$$ means twice the sum of length and width.
4. **Simplify inside the parentheses:**
$$3w + w = 4w$$
5. **Rewrite Expression 2 using distributive property:**
$$2(3w + w) = 2(4w) = 8w$$
6. **Simplify Expression 1:**
$$2(3w) + 2(w) = 6w + 2w = 8w$$
7. **Both expressions equal $$8w$$, showing the distributive property works.**
8. **For the regular hexagon with perimeter $$12c - 18$$:**
- Expression 1 for perimeter: $$12c - 18$$
- Expression 2 using distributive property: $$6(2c - 3)$$ since $$12c - 18 = 6(2c - 3)$$
9. **Find the length of one side:**
Since a regular hexagon has 6 equal sides, side length $$= \frac{12c - 18}{6} = 2c - 3$$
10. **Drawing a diagram helps visualize the hexagon and understand equal sides and perimeter division.**
**Final answers:**
- Expressions for perimeter: $$12c - 18$$ and $$6(2c - 3)$$
- Side length: $$2c - 3$$